There are several aspects of Charlotte Perkins Gilman's "An Obstacle" that make me believe that the narrator is female, however it is difficult for me to see these clues separate from what I know of Gilman as a person and a writer, in general:
She was a utopian feminist during a time when her accomplishments were exceptional for women, and she served as a role model for future generations of feminists because of her unorthodox concepts and lifestyle.
With this said, I feel the poem speaks for women, and the speaker is a woman. Support of this can be found by way of inference. The poem begins as the speaker notes she has a lot of responsibilities:
...With many things to do,
Important business of my own,
And other people's too...
I envision a woman who is not just a wife, but is also a parent and a child—perhaps even a friend, and/or servant to the needy; the things she has to accomplish are not just for herself, but in the capacity of all of these roles that she plays.
The "obstacle" the speaker confronts is "Prejudice." Note it is capitalized, which indicates not only its importance, but personifies it by giving it a proper name. (As the poem continues, the personification implied here is confirmed, especially in reference to "he.")
My strength and time were limited,
I carried quite a load...
This is similar to the old cliché, "...a woman's work is never done." Implied in literature, song, etc., one day for women often seems to hold more than what can be finished in twenty-four hours.
So I spoke to him politely,
For he was huge and high...
In the era in which Gilman lived, society demanded women follow a "code of conduct"—and so the speaker addresses "the obstacle" "politely." "Huge and wide" can refer to how far-reaching this prejudice is: with Gilman's proclivity towards writing feminist literature (see "The Yellow Wallpaper"), this obstacle permeates throughout every corner of society—seemingly insurmountable—putting limitations upon what a woman is allowed to do or not, e.g., the right to vote. When Prejudice smiles, likely it is done patronizingly—but he still won't budge.
In the fourth stanza, the woman (symbolic of womanhood) may be quietly losing her patience—for she uses reason (alluding to that of the Biblical Solomon), but "he" is a "fool" (rhyming with mule: an animal reputed to be stubborn and unreasonable).
And then I reasoned quietly
With that colossal mule...
I argued like a Solomon;
He sat there like a fool.
She is maintaining her composure (also expected of a woman of the time), "arguing" wisely, but to no avail. Then in the fifth stanza, we learn that she does lose her self-control, becoming angry...
...I flew into a passion,
and I danced and howled and swore.
He becomes equally angry. Can we not assume that he does so not just because he is reacting to her fury, but also because she is not acting "appropriately?"
In the next stanza, she is desperate: she begins to beg...one thing remaining for a woman (at that time), held against her will. Suddenly, inspiration comes. She realizes that had she not seen the better way, she would still be begging. However, she pulls herself together and...
...walked directly through him,
As if he wasn't there!
For she finally understands that she is not fighting flesh and blood, but simply popular opinion; and that by bowing to this, she cannot move. However, by ignoring it, her movement is no longer stymied and she is free to continue forward.